Chapter 1

Metric Spaces, Normed Spaces, Inner Product Spaces

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Section 1.1Basic Definitions and Theorems[10/0%]

Problem 1.1.1

(a)(b)(c)

(a)

Verify that $\left\| \cdot \right\|_{1},\left\| \cdot \right\|_{2}$ , and $\left\| \cdot \right\|_{\infty }$ define norms on $\mathbb {R}^{2}$ .

(b)

To see that these norms are in fact different, compute the distance $d((1,1),(2,3))$ (in each of the three norms) between the points $(1,1)$ and $(2,3)$ .

(c)

To see that norms are different, it helps (if it is possible) to visualize the "balls" in the space. Let

B_{r}(x)=\left\{ y \in V \mid \left\| x-y\right\| <r\right\} .

This is the open ball of radius $r$ centered at $x$ . Of special interest is the "unit ball," $B_{1}(0)=B_{1}((0,0))$ . In $\mathbb {R}^{2}$ , with the three different norms, sketch $B_{1}(0)$ and $B_{3}((2,2))$ .

Problem 1.1.2

(a)(b)

Consider the norms $\left\| \cdot \right\|_{1}$ and $\left\| \cdot \right\|_{\infty }$ on $\mathbb {R}^{n}$ .

(a)

Prove that

\left\| x\right\| =\frac{1}{3}\left\| x\right\| _{1}+\frac{2}{3}\left\| x\right\| _{\infty }

defines a norm on $\mathbb {R}^{n}$ .

(b)

Sketch the open unit ball in $\mathbb {R}^{2}$ with respect to this norm.

Problem 1.1.3

Verify that $\ell^{1}$ and $\ell^{\infty }$ are normed linear spaces.

Problem 1.1.4

Verify that $C([a, b])$ , with supremum norm, is a normed linear space.

Problem 1.1.5

Prove Theorem 1.1.

Theorem 1.1: Norms always give rise to metrics. Specifically, if $(V, \left\| \cdot \right\| )$ is a normed space and $d(v, w)$ is defined by $d(v, w) = \left\| v - w\right\|$ , then $d$ is a metric on $V$ .

Problem 1.1.6

Prove Theorem 1.2. (Hint: Use the Cauchy-Schwarz inequality to get the triangle inequality.)

Theorem 1.2: Inner products always give rise to norms. Specifically, if $(V, \left\langle \cdot , \cdot \right\rangle )$ is an inner product space and $\left\| v\right\|$ is defined by $\left\| v\right\| = \sqrt{ \left\langle v, v \right\rangle }$ , then $\left\| \cdot \right\|$ is a norm on $V$ .

Problem 1.1.7

Prove that in any complex inner product space

\langle v, \lambda w\rangle =\bar{\lambda }\langle v, w\rangle

for every $v, w \in V$ and each $\lambda \in \mathbb {C}$ .

Problem 1.1.8

(a)(b)

Consider a normed linear space $(V,\left\| \cdot \right\| )$ . Recall that some normed linear spaces are inner product spaces, and some are not.

(a)

Prove that the parallelogram equality characterizes the inner product spaces among the normed linear spaces. That is, show that

\left\| v\right\| =\sqrt{\langle v, v\rangle }

for some inner product $\langle \cdot , \cdot \rangle$ if and only if

2\left\| u\right\| ^{2}+2\left\| v\right\| ^{2}=\left\| u+v\right\| ^{2}+\left\| u-v\right\| ^{2}

holds for every pair $u$ and $v$ in $V$ .

(b)

Is the parallelogram equality satisfied in $\ell^{1}$ ? In $\ell^{\infty }$ ?

Problem 1.1.9

Use the preceding exercise to show that the supremum norm on $C([a, b])$ cannot come from an inner product.

Problem 1.1.10

(a)(b)

(a)

In $C([0,1])$ with supremum norm, compute $d(f, g)$ for $f(x)=1$ and $g(x)=x$ .

(b)

Repeat part (a), with the supremum norm replaced by the norm induced by the inner product

\langle f, g\rangle =\int _{0}^{1} f(x) g(x) d x

Section 1.3A Discussion About Dimension[3/0%]

Problem 1.3.1

The basis for $\mathbb {R}^{n}$ given in the text is called the "standard basis." It is a basis, but not the only basis, for $\mathbb {R}^{n}$ . Give another basis for $\mathbb {R}^{3}$ , and show that it is in fact a basis.

Problem 1.3.2

Prove that $\ell^{1}$ is infinite-dimensional. Explain why your proof also shows that $\ell^{\infty }$ is infinite-dimensional.

Problem 1.3.3

Prove that $C([0,1])$ is infinite-dimensional.